Suppose we have two known continuous functions $g$ and $h$, and want to make another, preferably continuous, function $f$ such that $f(x<a)=g(x)$ and $f(x>b)=h(x)$. In other words, in the range $[a,b]$, $f$ undergoes a transition from $g$ to $h$. A suitable form for $f(x)$ has to be chosen in the range $[a,b]$, for example
$$f(x)=\frac{(x-a)^2h(x)+(x-b)^2g(x)}{(x-a)^2+(x-b)^2}$$
As a physicist not particularly familiar with mathematical terminology, especially in English, I would like to know if the described process of making/choosing $f$ could be called "interpolation of functions $g$ and $h$", and what forms are commonly used by mathematicians for $f$ in the transition range.
You are really constructing the function from 3 branches, the only interpolation in your example is done on $(a,b)$.
As for the forms to use, the most basic is linear interpolation, which guarantees continuity but unfortunately does not make the result differentiable.
To achieve differentiability (assuming $g'(a)$ and $h'(b)$ are well-defined) you can use something like a cubic, for example, since you now are imposing 4 conditions, 2 on the function and 2 on its derivative.
If you have a series of such points, then a spline could be a good idea.